1. Field of the Invention
This invention relates to geometric surfaces, and more particularly to a model for illustrating the geometric properties of the hyperbolic plane and to a structure for providing a sheet of thin, flexible material which cannot be flattened and which has a large surface area in relation to its volume.
2. Description of the Prior Art
Three basic geometrical surfaces are the Euclidean plane, the spherical plane and the hyperbolic plane. Each of these planes have their own geometric properties. For example, in the familiar Euclidean plane a pair of lines intersecting a third line at right angles are parallel to each other. In a spherical plane, a pair of lines intersecting a third line at right angles (e.g. a pair of meridians intersecting the equator) converge toward each other and eventually overlap. In the hyperbolic plane, a pair of lines intersecting a third line at right angles continuously diverge away from each other.
The geometric properties of the Euclidean plane can be easily demonstrated on a flat, planar surface. Similarly, a spherical geometry model for illustrating the properties of spherical geometry is formed by a simple sphere. However, it has not heretofore been possible to produce a physical model for demonstrating the properties of hyperbolic geometry, except on patches of small intrinsic size.
Although physical, hyperbolic geometry models have not been produced, the characteristics of the hyperbolic plane have been extensively studied and are described in detail in Coxeter, H. S. M., Non-Euclidean Geometry, pages 147-167, University of Toronto Press, Toronto, Canada 1942 and Coxeter, H. S. M., Twelve Geometric Essays, pages 199-214). Southern Illinois University Press, Carbondale, Illinois 1968. These references decompose the hyperbolic plane into "tessellations" or "tilings"; that is, an arrangement of points, line-segments and simple polygons (called vertices, edges and faces, respectively) such that every edge joins two of the vertices and it is a common side of two of the faces. Although the hyperbolic plane can be described as being composed of these tessellations, the tessellations disclosed therein are distorted so that they can be illustrated in a Euclidean plane. This distortion is necessary since the sum of the angles of the corresponding Euclidean polygons about each vertex exceeds 360.degree., and it is not possible to accurately portray a collection of polygons having this property in the Euclidean plane.